Kinematic Equations | QL #76
What kinematic equations can be used to solve for displacement, velocity, acceleration, and time?
Kinematic Equation for Velocity as a Function of Time
Imagine an object undergoing constant acceleration, and consider the equation of
motion, v = v0 + at.
1. Show that the equation above is a simple restatement of the definition of acceleration, a =
Δv/Δt. Explain all steps.
a = Δv/δt = (v - v0)/t, where it is assumed that t0 = 0 and the final velocity is represented
as v. Solving for v yields v = v0 + at.
2. Suppose that, initially, the object is moving to the right at 40 m/s. If the object
is not accelerating, what will be its velocity at time t = 4 seconds?Explain.
Its velocity is still 40 m/s because zero acceleration means "no change in the velocity."
3. Suppose the object is accelerating, such that its velocity increases with time at 7 m/s2. In
which direction does its acceleration vector point?
It also points to the right, because the velocity is increasing and the velocity vector
points to the right.
4. Using v = v0 + at, show that the object's velocity, after four seconds, will be 68 m/s to the
right.
v = v0 + at = +40 + 7(4) = 40 + 28 = 68 m/s to the right.
5. In Table 1, the same problem is considered but with different values for the initial velocity and
acceleration. (The first table row displays the results of problem 4.) In the second row, the
initial velocity of the object is still 40 m/s to the right, but the object is accelerating to the
left (that is, in the negative direction) at 7 m/s2. Fill in the rest of the table, using v = v0 + at.
Table 1
v0 (m/s)
a (m/s )
v* (m/s)
Δv (m/s)
at (m/s)
+40
+7
+68
+28
+28
+40
-7
+12
-28
-28
+20
+7
+48
+28
+28
+20
-7
-8
-28
-28
* Denotes the velocity of the object after t = 4 seconds.
2
Kinematic Equation for Displacement as a Function of Time
Now, consider the next kinematic equation for linear motion, d = v0t + (1/2)at2.
6. Why are the two terms, v0t and (1/2)at2, vectors?
Because time is a scalar quantity, and the product of a vector and any number of
scalar quantities is still a vector.
7. Suppose an object starts out moving to the right at 4 m/s. If the object is not accelerating,
explain why its displacement after t = 4 seconds will be 16 meters to the right. (This distance
that is traveled with no acceleration will be called d.)
With a = 0, the displacement becomes d = v0t = +4(4) = 16 meters to the right.
8. Suppose the object is accelerating such that its velocity increases with time at 1 m/s2. After
four seconds, explain why its displacement, d, is 24 meters to the right. (Notice that this
acceleration produces an increase in displacement, Δd, of 8 meters to the right.)
d = v0t + (1/2)at2 = +4(4) + (1/2)(1)(4)2 = 16 + 8 = 24 meters to the right.
9. Fill in Table 2 for various initial velocities and accelerations. The first row has been completed
for you.
Table 2
d*
v0t
d**
Δd
(1/2)at2
v0
a
+4
+16
+16
+1
+24
+8
+8
+4
-1
+16
+16
+8
-8
-8
+2
+1
+8
+8
+16
+8
+8
+2
-1
+8
+8
0
-8
-8
* Denotes the displacement that would occur if a = 0.
** Denotes the displacement of the accelerating object after t = 4 s.
10. For the first three columns in each row of Table 1, explain what is occurring physically. An
example is provided below:
In the first trial, the object is initially traveling to the right at 40 m/s, and is increasing in speed
at a rate of 7 m/s2. This means its acceleration vector points to the right as well. After four
seconds, the object is traveling at 68 m/s to the right.
Answers will vary.
11. By examining the last two columns in Table 1, summarize the physical significance of
the at term in the equation v = v0 + at. How does the directionof the vector at affect your
answer?
If an object is not accelerating, it will maintain its original speed, v0. The at term in the
equation represents the change in velocity that occurs because the object is
accelerating. When at points in the same direction as v0, it tells you how much extra
velocity is gained over its initial velocity, v0. When at points in the opposite direction
of v0, the at term tells you by how much the final velocity vector is lower than v0.
12. From Table 2, summarize the physical significance of the v0t and (1/2)at2 terms in the
equation d = v0t + (1/2)at2. How does the direction of the acceleration vector, a, affect your
answer?
The v0t term is the displacement of an object if it is not accelerating. However, if an
object is accelerating, you would not expect it to have the same displacement. For
example, if the object were speeding up, it would be expected to travel farther, and if it
were slowing down, it would be expected to travel less far. The (1/2)at2 term indicates
how much the displacement changes due to the acceleration of the object.
If (1/2)at2 points in the same direction as v0, this term indicates how much greater the
displacement becomes. If (1/2)at2 points in the opposite direction, this term indicates
how much less the displacement becomes.